Geodesic
On Stieltjes integrals and Parseval's equality for multiple trigonometric series
Izvestiya. Mathematics , Tome 69 (2005) no. 5, pp. 1005-1024.

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In this paper, it is proved that if a function $f$ from $\mathbb R^n$ to $\mathbb C$ is $2\pi$-periodic with respect to each variable and Lebesgue integrable on $T^n=[0,2\pi]^n$, a complex-valued additive segment function $\mathcal G$ is defined on all segments in $\mathbb R^n$ and is $2\pi$-periodic with respect to each variable, the point function $G$ corresponding to $\mathcal G$ is Lebesgue integrable on $T^n$, and the function $f$ is integrable with respect to $\overline{\mathcal G}$ in the Riemann–Stieltjes sense on all shifts of $T^n$, then Parseval's equality holds with the series not necessarily convergent, but summable by Riemann's method. Some results are also obtained on Parseval's equality for Fourier–Lebesgue–Stieltjes multiple trigonometric series. 
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     title = {On {Stieltjes} integrals and {Parseval's} equality for multiple trigonometric series},
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T. P. Lukashenko. On Stieltjes integrals and Parseval's equality for multiple trigonometric series. Izvestiya. Mathematics , Tome 69 (2005) no. 5, pp. 1005-1024. http://geodesic.mathdoc.fr/item/IM2_2005_69_5_a4/

[1] Zigmund A., Trigonometricheskie ryady, T. 1, Mir, M., 1965 | MR

[2] Goryacheva V. S., “O ravenstve Parsevalya dlya ryadov Fure–Stiltesa”, Vestn. Mosk. un-ta. Matem. Mekhan., 2002, no. 2, 32–36 | MR | Zbl

[3] Lukashenko T. P., “Ob integrale Rimana–Stiltesa i ravenstve Parsevalya”, Vestn. Mosk. un-ta. Matem. Mekhan., 2002, no. 4, 18–23 | MR | Zbl

[4] Ter-Krikorov A. M., Shabunin M. I., Kurs matematicheskogo analiza, Nauka, M., 1988 | MR | Zbl

[5] Saks S., Teoriya integrala, Faktorial Press, M., 2004

[6] Edvards R., Ryady Fure v sovremennom izlozhenii, T. 1, Mir, M., 1985

[7] Bari N. K., Trigonometricheskie ryady, Fizmatgiz, M., 1961 | MR

[8] Henstock R., “Definitions of Riemann type of the variational integrals”, Proc. London Math. Soc., 11(43) (1961), 402–418 | DOI | MR | Zbl

[9] Zigmund A., Trigonometricheskie ryady, T. 2, Mir, M., 1965 | MR

[10] Zhizhiashvili L. V., Nekotorye voprosy mnogomernogo garmonicheskogo analiza, Izd-vo Tbilisskogo un-ta, Tbilisi, 1983 | MR | Zbl

[11] Golubov B. I., “Kratnye ryady i integraly Fure”, Itogi nauki i tekhniki. Matem. analiz, 19, VINITI, M., 1982, 3–54 | MR

[12] Tolstov G. P., Mera i integral, Nauka, M., 1976 | MR | Zbl

[13] Tolstov G. P., “Zamechanie k teoreme D. F. Egorova”, DAN SSSR, 22 (1939), 309–311